Questions D2 (547 questions)

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AQA D2 2014 June Q3
3 The diagram below shows a network of pipes with source \(A\) and \(\operatorname { sink } J\). The capacity of each pipe is given by the number on each edge.
\includegraphics[max width=\textwidth, alt={}, center]{c2b62fee-d320-4701-a5bb-b2e4b8cc0952-08_816_1280_443_386}
  1. Find the values of the cuts \(\mathrm { C } _ { 1 }\) and \(\mathrm { C } _ { 2 }\).
  2. Find by inspection a flow of 60 units, with flows of 25,10 and 25 along \(H J , G J\) and \(I J\) respectively. Illustrate your answer on Figure 1.
    1. On a certain day the section \(E H\) is blocked, as shown on Figure 2. Find, by inspection or otherwise, the maximum flow on this day and illustrate your answer on Figure 2.
    2. Show that the flow obtained in part (c)(i) is maximal. \begin{figure}[h]
      \captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{c2b62fee-d320-4701-a5bb-b2e4b8cc0952-09_595_1065_376_475}
      \end{figure}
  4. \begin{figure}[h]
    \captionsetup{labelformat=empty} \caption{Figure 2} \includegraphics[alt={},max width=\textwidth]{c2b62fee-d320-4701-a5bb-b2e4b8cc0952-09_617_1061_1142_477}
    \end{figure} Maximum flow = \(\_\_\_\_\)
AQA D2 2014 June Q4
4
  1. Display the following linear programming problem in a Simplex tableau. $$\begin{array} { l c } \text { Maximise } & P = 3 x + 6 y + 2 z
    \text { subject to } & x + 3 y + 2 z \leqslant 11
    & 3 x + 4 y + 2 z \leqslant 21
    \text { and } & x \geqslant 0 , y \geqslant 0 , z \geqslant 0 . \end{array}$$
  2. The first pivot to be chosen is from the \(y\)-column. Perform one iteration of the Simplex method.
  3. Perform one further iteration.
  4. Interpret the tableau obtained in part (c) and state the values of your slack variables.
AQA D2 2014 June Q5
7 marks
5 Mark and Owen play a zero-sum game. The game is represented by the following pay-off matrix for Mark.
Owen
\cline { 2 - 5 }\cline { 2 - 5 }StrategyDEF
A41- 1
\cline { 2 - 5 } MarkB3- 2- 2
\cline { 2 - 5 }C- 203
  1. Explain why Mark should never play strategy B.
  2. It is given that the value of the game is 0.6 . Find the optimal strategy for Owen.
    (You are not required to find the optimal mixed strategy for Mark.)
    [0pt] [7 marks]
AQA D2 2014 June Q6
12 marks
6 The network below has 11 vertices and 16 edges connecting some pairs of vertices. The numbers on the edges are their weights. The weight of the edge \(D G\) is given in terms of \(x\). There are three routes from \(A\) to \(K\) that have the same minimum total weight.
\includegraphics[max width=\textwidth, alt={}, center]{c2b62fee-d320-4701-a5bb-b2e4b8cc0952-16_863_1444_552_299} Working backwards from \(\boldsymbol { K }\), use dynamic programming, to find:
  1. the minimum total weight from \(A\) to \(K\);
  2. the value of \(x\);
  3. the three routes corresponding to the minimum total weight. You must complete the table opposite as your solution.
    [0pt] [12 marks] \section*{Answer space for question 6}
    StageStateFromCalculationValue
    1IK
    \(J\)K
AQA D2 2014 June Q7
7 The table shows the times taken, in minutes, by four people, \(A , B , C\) and \(D\), to carry out the tasks \(W , X , Y\) and \(Z\). Some of the times are subject to the same delay of \(x\) minutes, where \(4 < x < 11\).
AQA D2 2014 June Q8
8 An activity diagram for a project is shown below. The duration of each activity is given in weeks. The earliest start time and the latest finish time for each activity are shown on the diagram.
\includegraphics[max width=\textwidth, alt={}, center]{c2b62fee-d320-4701-a5bb-b2e4b8cc0952-22_640_1626_475_209}
  1. Find the values of \(x , y\) and \(z\).
  2. State the critical path.
  3. Some of the activities can be speeded up at an additional cost. The following table lists the activities that can be speeded up together with the minimum possible duration of these activities. The table also shows the additional cost of reducing the duration of each of these activities by one week.
AQA D2 2015 June Q1
2 marks
1 Figure 2, on the page opposite, shows an activity diagram for a project. Each activity requires one worker. The duration required for each activity is given in hours.
  1. On Figure 1 below, complete the precedence table.
  2. Find the earliest start time and the latest finish time for each activity and insert their values on Figure 2.
  3. List the critical paths.
  4. Find the float time of activity \(E\).
  5. Using Figure 3 opposite, draw a Gantt diagram to illustrate how the project can be completed in the minimum time, assuming that each activity is to start as early as possible.
  6. Given that there is only one worker available for the project, find the minimum completion time for the project.
  7. Given that there are two workers available for the project, find the minimum completion time for the project. Show a suitable allocation of tasks to the two workers.
    [0pt] [2 marks] \begin{table}[h]
    \captionsetup{labelformat=empty} \caption{Figure 1}
    ActivityImmediate predecessor(s)
    A
    B
    C
    D
    E
    \(F\)
    G
    \(H\)
    I
    J
    \end{table} \begin{figure}[h]
    \captionsetup{labelformat=empty} \caption{Figure 2} \includegraphics[alt={},max width=\textwidth]{b0f9523e-51dd-495f-99ec-4724243b5619-03_1071_1561_376_278}
    \end{figure} \begin{figure}[h]
    \captionsetup{labelformat=empty} \caption{Figure 3} \includegraphics[alt={},max width=\textwidth]{b0f9523e-51dd-495f-99ec-4724243b5619-03_801_1301_1644_420}
    \end{figure}
AQA D2 2015 June Q2
2 Stan and Christine play a zero-sum game. The game is represented by the following pay-off matrix for Stan. \begin{table}[h]
\captionsetup{labelformat=empty} \caption{Christine}
\multirow{5}{*}{Stan}StrategyDEF
A3- 3- 1
B- 1- 42
C10- 3
\cline { 2 - 5 }- 2
\end{table}
  1. Find the play-safe strategy for each player.
  2. Show that there is no stable solution.
  3. Explain why a suitable pay-off matrix for Christine is given by
AQA D2 2015 June Q3
11 marks
3 In the London 2012 Olympics, the Jamaican \(4 \times 100\) metres relay team set a world record time of 36.84 seconds. Athletes take different times to run each of the four legs.
The coach of a national athletics team has five athletes available for a major championship. The lowest times that the five athletes take to cover each of the four legs is given in the table below. The coach is to allocate a different athlete from the five available athletes, \(A , B , C , D\) and \(E\), to each of the four legs to produce the lowest total time.
Leg 1Leg 2Leg 3Leg 4
Athlete \(\boldsymbol { A }\)9.848.918.988.70
Athlete \(\boldsymbol { B }\)10.289.069.249.05
Athlete \(\boldsymbol { C }\)10.319.119.229.18
Athlete \(\boldsymbol { D }\)10.049.079.199.01
Athlete \(\boldsymbol { E }\)9.918.959.098.74
Use the Hungarian algorithm, by reducing the columns first, to assign an athlete to each leg so that the total time of the four athletes is minimised. State the allocation of the athletes to the four legs and the total time.
[0pt] [11 marks]
\includegraphics[max width=\textwidth, alt={}]{b0f9523e-51dd-495f-99ec-4724243b5619-08_1200_1705_1507_155}
AQA D2 2015 June Q4
4
  1. Display the following linear programming problem in a Simplex tableau. $$\begin{array} { l r } \text { Maximise } & P = 2 x + 3 y + 4 z
    \text { subject to } & x + y + 2 z \leqslant 20
    & 3 x + 2 y + z \leqslant 30
    & 2 x + 3 y + z \leqslant 40
    \text { and } & x \geqslant 0 , y \geqslant 0 , z \geqslant 0 \end{array}$$
    1. The first pivot to be chosen is from the \(z\)-column. Identify the pivot and explain why this particular value is chosen.
    2. Perform one iteration of the Simplex method.
    1. Perform one further iteration.
    2. Interpret your final tableau and state the values of your slack variables.
AQA D2 2015 June Q5
2 marks
5 Tom is going on a driving holiday and wishes to drive from \(A\) to \(K\).
The network below shows a system of roads. The number on each edge represents the maximum altitude of the road, in hundreds of metres above sea level. Tom wants to ensure that the maximum altitude of any road along the route from \(A\) to \(K\) is minimised.
\includegraphics[max width=\textwidth, alt={}, center]{b0f9523e-51dd-495f-99ec-4724243b5619-14_1522_1363_660_342}
  1. Working backwards from \(\boldsymbol { K }\), use dynamic programming to find the optimal route when driving from \(A\) to \(K\). You must complete the table opposite as your solution.
  2. Tom finds that the road \(C F\) is blocked. Find Tom's new optimal route and the maximum altitude of any road on this route.
    [0pt] [2 marks] \section*{Answer space for question 5}
    StageStateFromValue
    1H\(K\)
    I\(K\)
    \(J\)\(K\)
    2
  3. Optimal route is \(\_\_\_\_\)
  4. Tom's route is \(\_\_\_\_\)
    Maximum altitude is \(\_\_\_\_\) Figure 4 below shows a network of pipes.
    The capacity of each pipe is given by the number not circled on each edge. The numbers in circles represent an initial flow. \begin{figure}[h]
    \captionsetup{labelformat=empty} \caption{Figure 4} \includegraphics[alt={},max width=\textwidth]{b0f9523e-51dd-495f-99ec-4724243b5619-18_1039_1623_561_191}
    \end{figure}
  5. Find the value of the initial flow.
    1. Use the initial flow and the labelling procedure on Figure 5 to find the maximum flow through the network. You should indicate any flow-augmenting routes in the table and modify the potential increases and decreases of the flow on the network.
    2. State the value of the maximum flow and, on Figure 6, illustrate a possible flow along each edge corresponding to this maximum flow.
  7. Confirm that you have a maximum flow by finding a cut of the same value. List the edges of your cut.
  8. On a particular day, there is a restriction at vertex \(G\) which allows a maximum flow through \(G\) of 30 . Find, by inspection, the maximum flow through the network on this day.
  9. Initial flow \(=\) \(\_\_\_\_\)
    1. Figure 5
      \includegraphics[max width=\textwidth, alt={}, center]{b0f9523e-51dd-495f-99ec-4724243b5619-19_2158_1559_543_296}
      \(7 \quad\) Arsene and Jose play a zero-sum game. The game is represented by the following pay-off matrix for Arsene, where \(x\) is a constant. The value of the game is 2.5 .
      Jose
      \cline { 2 - 4 }StrategyCD
      \cline { 2 - 4 } ArseneA\(x + 3\)1
      \cline { 2 - 4 }B\(x + 1\)3
      \cline { 2 - 4 }
      \cline { 2 - 4 }
  11. Find the optimal mixed strategy for Arsene.
  12. Find the value of \(x\).
    \includegraphics[max width=\textwidth, alt={}]{b0f9523e-51dd-495f-99ec-4724243b5619-22_1636_1707_1071_153}
    \includegraphics[max width=\textwidth, alt={}]{b0f9523e-51dd-495f-99ec-4724243b5619-24_2288_1705_221_155}
AQA D2 2016 June Q1
1
Figure 1 below shows an activity diagram for a project. Each activity requires one worker. The duration required for each activity is given in hours.
  1. Find the earliest start time and the latest finish time for each activity and insert these values on Figure 1.
    1. Find the critical path.
    2. Find the float time of activity \(F\).
  3. Using Figure 2 on page 3, draw a resource histogram to illustrate how the project can be completed in the minimum time, assuming that each activity is to start as early as possible.
    1. Given that there are two workers available for the project, find the minimum completion time for the project.
    2. Write down an allocation of tasks to the two workers that corresponds to your answer in part (d)(i). \section*{Answer space for question 1} \begin{figure}[h]
      \captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{34de3f03-a275-44fb-88b2-b88038bcec97-02_687_1655_1941_189}
      \end{figure} \section*{Answer space for question 1} \begin{figure}[h]
      \captionsetup{labelformat=empty} \caption{Figure 2} \includegraphics[alt={},max width=\textwidth]{34de3f03-a275-44fb-88b2-b88038bcec97-03_1115_1575_434_283}
      \end{figure}
      \includegraphics[max width=\textwidth, alt={}]{34de3f03-a275-44fb-88b2-b88038bcec97-03_1024_1593_1683_267}
AQA D2 2016 June Q2
2 Alan, Beth, Callum, Diane and Ethan work for a restaurant chain. The costs, in pounds, for the five people to travel to each of five different restaurants are recorded in the table below. Alan cannot travel to restaurant 1 and Beth cannot travel to restaurants 3 and 5, as indicated by the asterisks in the table.
AQA D2 2016 June Q3
3 marks
3
Maximise \(\quad P = 2 x - 3 y + 4 z\)
subject to \(\quad x + 2 y + z \leqslant 20\)
\(x - y + 3 z \leqslant 24\)
\(3 x - 2 y + 2 z \leqslant 30\)
and \(\quad x \geqslant 0 , y \geqslant 0 , z \geqslant 0\).
  1. Display the linear programming problem in a Simplex tableau.
    1. The first pivot to be chosen is from the \(z\)-column. Identify the pivot and explain why this particular value is chosen.
    2. Perform one iteration of the Simplex method.
    3. Perform one further iteration.
  3. Interpret your final tableau and state the values of your slack variables.
    [0pt] [3 marks]
AQA D2 2016 June Q4
4 Monica and Vladimir play a zero-sum game. The game is represented by the following pay-off matrix for Monica.
AQA D2 2016 June Q5
5 Robert is planning to renovate four houses, \(A , B , C\) and \(D\), at the rate of one per month. The houses can be renovated in any order but the costs will vary because some of the materials left over from renovating one house can be used for the next one. The expected profits, in hundreds of pounds, are given in the table below.
AQA D2 2016 June Q6
6 The network shows a system of pipes with lower and upper capacities for each pipe in litres per second.
\includegraphics[max width=\textwidth, alt={}, center]{34de3f03-a275-44fb-88b2-b88038bcec97-22_817_744_397_648}
    1. Find the value of the cut \(X\).
    2. Hence state what can be deduced about the maximum flow from \(A\) to \(H\).
  2. Figure 3 shows a partially completed diagram for a feasible flow of 28 litres per second from \(A\) to \(H\). Indicate, on Figure 3, the flows along the edges \(B D , B E\) and \(C D\).
    1. Using your feasible flow from part (b) as an initial flow, indicate potential increases and decreases of the flow along each edge on Figure 4.
    2. Use flow augmentation on Figure 4 to find the maximum flow from \(A\) to \(H\). You should indicate any flow augmenting paths in the table and modify the potential increases and decreases of the flow on the network.
    3. State the maximum flow and indicate a maximum flow on Figure 5. \section*{Answer space for question 6} \begin{figure}[h]
      \captionsetup{labelformat=empty} \caption{Figure 3} \includegraphics[alt={},max width=\textwidth]{34de3f03-a275-44fb-88b2-b88038bcec97-23_682_689_312_397}
      \end{figure} \begin{figure}[h]
      \captionsetup{labelformat=empty} \caption{Figure 4} \includegraphics[alt={},max width=\textwidth]{34de3f03-a275-44fb-88b2-b88038bcec97-23_935_1477_1037_365}
      \end{figure} Figure 5
      \includegraphics[max width=\textwidth, alt={}]{34de3f03-a275-44fb-88b2-b88038bcec97-24_2032_1707_219_153}
Edexcel D2 Q1
1. \section*{Figure 1}
\includegraphics[max width=\textwidth, alt={}]{4f494f19-5690-4d9f-8c18-db03d41da203-01_435_682_383_374}
Figure 1 shows a network of roads connecting six villages \(A , B , C , D , E\) and \(F\). The lengths of the roads are given in km.
  1. Complete the table in the answer booklet, in which the entries are the shortest distances between pairs of villages. You should do this by inspection.
    (2) The table can now be taken to represent a complete network.
  2. Use the nearest-neighbour algorithm, starting at \(A\), on your completed table in part (a). Obtain an upper bound to the length of a tour in this complete network, which starts and finishes at \(A\) and visits every village exactly once.
  3. Interpret your answer in part (b) in terms of the original network of roads connecting the six villages.
  4. By choosing a different vertex as your starting point, use the nearest-neighbour algorithm to obtain a shorter tour than that found in part (b). State the tour and its length.
Edexcel D2 Q2
2. A two-person zero-sum game is represented by the following pay-off matrix for player \(A\).
\(B\)
IIIIIIIV
\cline { 2 - 6 }I- 4- 5- 24
AII- 11- 12
III05- 2- 4
IV- 13- 11
  1. Determine the play-safe strategy for each player.
  2. Verify that there is a stable solution and determine the saddle points.
  3. State the value of the game to \(B\). Figure 2 \begin{figure}[h]
    \captionsetup{labelformat=empty} \caption{3.} \includegraphics[alt={},max width=\textwidth]{4f494f19-5690-4d9f-8c18-db03d41da203-02_502_990_319_148}
    \end{figure} The network in Fig. 2 shows possible routes that an aircraft can take from \(S\) to \(T\). The numbers on the directed arcs give the amount of fuel used on that part of the route, in appropriate units. The airline wishes to choose the route for which the maximum amount of fuel used on any part of the route is as small as possible. This is the rninimax route.
  4. Complete the table in the answer booklet.
  5. Hence obtain the minimax route from \(S\) to \(T\) and state the maximum amount of fuel used on any part of this route.
    (2)
    4. Andrew ( \(A\) ) and Barbara ( \(B\) ) play a zero-sum game. This game is represented by the following payoff matrix for Andrew. $$A \left( \begin{array} { c c c } & B &
    3 & 5 & 4
    1 & 4 & 2
    6 & 3 & 7 \end{array} \right)$$
  6. Explain why this matrix may be reduced to $$\left( \begin{array} { l l } 3 & 5
    6 & 3 \end{array} \right) .$$
  7. Hence find the best strategy for each player and the value of the game.
    5. An engineering company has 4 machines available and 4 jobs to be completed. Each machine is to be assigned to one job. The time, in hours, required by each machine to complete each job is shown in the table below.
    \cline { 2 - 5 } \multicolumn{1}{c|}{}Job 1Job 2Job 3Job 4
    Machine 114587
    Machine 221265
    Machine 37839
    Machine 424610
    Use the Hungarian algorithm, reducing rows first, to obtain the allocation of machines to jobs which minimises the total time required. State this minimum time.
    6. The table below shows the distances, in km , between six towns \(A , B , C , D , E\) and \(F\).
    \cline { 2 - 7 } \multicolumn{1}{c|}{}\(A\)\(B\)\(C\)\(D\)\(E\)\(F\)
    \(A\)-85110175108100
    \(B\)85-3817516093
    \(C\)11038-14815673
    \(D\)175175148-11084
    \(E\)108160156110-92
    \(F\)10093738492-
  8. Starting from \(A\), use Prim's algorithm to find a minimum connector and draw the minimum spanning tree. You must make your method clear by stating the order in which the arcs are selected.
    1. Using your answer to part (a) obtain an initial upper hound for the solution of the travelling salesman problem.
    2. Use a short cut to reduce the upper bound to a value less than 680 .
  10. Starting by deleting \(F\), find a lower bound for the solution of the travelling salesman problem.
    7. A steel manufacturer has 3 factories \(F _ { 1 } , F _ { 2 }\) and \(F _ { 3 }\) which can produce 35,25 and 15 kilotonnes of steel per year, respectively. Three businesses \(B _ { 1 } , B _ { 2 }\) and \(B _ { 3 }\) have annual requirements of 20,25 and 30 kilotonnes respectively. The table below shows the cost \(C _ { i j }\) in appropriate units, of transporting one kilotonne of steel from factory \(F _ { i }\) to business \(B _ { j }\).
    \cline { 3 - 5 } \multicolumn{2}{c|}{}Business
    \cline { 3 - 5 } \multicolumn{2}{c|}{}\(B _ { 1 }\)\(B _ { 2 }\)\(B _ { 3 }\)
    \multirow{3}{*}{Factory}\(F _ { 1 }\)10411
    \cline { 2 - 5 }\(F _ { 2 }\)1258
    \cline { 2 - 5 }\(F _ { 3 }\)967
    The manufacturer wishes to transport the steel to the businesses at minimum total cost.
  11. Write down the transportation pattern obtained by using the North-West corner rule.
  12. Calculate all of the improvement indices \(I _ { i j }\), and hence show that this pattern is not optimal.
  13. Use the stepping-stone method to obtain an improved solution.
  14. Show that the transportation pattern obtained in part (c) is optimal and find its cost. \begin{figure}[h]
    \captionsetup{labelformat=empty} \caption{Figure 4} \includegraphics[alt={},max width=\textwidth]{4f494f19-5690-4d9f-8c18-db03d41da203-04_346_922_319_278}
    \end{figure} The network in Fig. 4 models a drainage system. The number on each arc indicates the capacity of that arc, in litres per second.
  15. Write down the source vertices. \begin{figure}[h]
    \captionsetup{labelformat=empty} \caption{Figure 5} \includegraphics[alt={},max width=\textwidth]{4f494f19-5690-4d9f-8c18-db03d41da203-04_341_920_900_278}
    \end{figure} Figure 5 shows a feasible flow through the same network.
  16. State the value of the feasible flow shown in Fig. 5. Taking the flow in Fig. 5 as your initial flow pattern,
  17. use the labelling procedure on Diagram 1 to find a maximum flow through this network. You should list each flow-augmenting route you use, together with its flow.
  18. Show the maximal flow on Diagram 2 and state its value.
  19. Prove that your flow is maximal.
    9. T42 Co. Ltd produces three different blends of tea, Morning, Afternoon and Evening. The teas must be processed, blended and then packed for distribution. The table below shows the time taken, in hours, for each stage of the production of a tonne of tea. It also shows the profit, in hundreds of pounds, on each tonne.
    \cline { 2 - 5 } \multicolumn{1}{c|}{}ProcessingBlendingPackingProfit (£100)
    Morning blend3124
    Afternoon blend2345
    Evening blend4233
    The total times available each week for processing, blending and packing are 35, 20 and 24 hours respectively. T42 Co. Ltd wishes to maximise the weekly profit. Let \(x , y\) and \(z\) be the number of tonnes of Morning, Afternoon and Evening blend produced each week.
  20. Formulate the above situation as a linear programming problem, listing clearly the objective function, and the constraints as inequalities. An initial Simplex tableau for the above situation is
    Basic
    variable
    		\(x\)\(y\)\(z\)\(r\)\(s\)\(t\)Value
    \(r\)32410035
    \(s\)13201020
    \(t\)24300124
    \(P\)- 4- 5- 30000
  21. Solve this linear programming problem using the Simplex algorithm. Take the most negative number in the profit row to indicate the pivot column at each stage.
    (11) T42 Co. Ltd wishes to increase its profit further and is prepared to increase the time available for processing or blending or packing or any two of these three.
  22. Use your answer to part (b) to advise the company as to which stage(s) it should increase the time available.
    (2)
    10. While solving a maximizing linear programming problem, the following tableau was obtained.
    Basic
    variable
    		\(x\)\(y\)\(z\)\(r\)\(s\)\(t\)Value
    \(r\)00\(1 \frac { 2 } { 3 }\)10\(- \frac { 1 } { 6 }\)\(\frac { 2 } { 3 }\)
    \(y\)01\(3 \frac { 1 } { 3 }\)01\(- \frac { 1 } { 3 }\)\(\frac { 1 } { 3 }\)
    \(x\)10- 30- 1\(\frac { 1 } { 2 }\)1
    \(P\)00101111
  23. Explain why this is an optimal tableau.
  24. Write down the optimal solution of this problem, stating the value of every variable.
  25. Write down the profit equation from the tableau. Use it to explain why changing the value of any of the non-basic variables will decrease the value of \(P\).
    11. A company wishes to transport its products from 3 factories \(F _ { 1 } , F _ { 2 }\) and \(F _ { 3 }\) to a single retail outlet \(R\). The capacities of the possible routes, in van loads per day, are shown in Fig. 5. \begin{figure}[h]
    \captionsetup{labelformat=empty} \caption{Figure 5} \includegraphics[alt={},max width=\textwidth]{4f494f19-5690-4d9f-8c18-db03d41da203-05_470_766_447_1695}
    \end{figure}
  26. On Diagram 1 in the answer booklet add a supersource \(S\) to obtain a capacitated network with a single source and a single sink. State the minimum capacity of each arc you have added.
    1. State the maximum flow along \(S F _ { 1 } A B R\) and \(S F _ { 3 } C R\).
    2. Show these maximum flows on Diagram 2 in the answer booklet, using numbers in circles. Taking your answer to part (b)(ii) as the initial flow pattern,
    1. use the labelling procedure to find a maximum flow from \(S\) to \(R\). Your working should be shown on Diagram 3. List each flow-augmenting route you find together with its flow.
    2. Prove that your final flow is maximal.
      12. \begin{figure}[h]
      \captionsetup{labelformat=empty} \caption{Figure 2} \includegraphics[alt={},max width=\textwidth]{4f494f19-5690-4d9f-8c18-db03d41da203-06_405_791_301_221}
      \end{figure} A company has 3 warehouses \(W _ { 1 } , W _ { 2 }\), and \(W _ { 3 }\). It needs to transport the goods stored there to 2 retail outlets \(R _ { 1 }\) and \(R _ { 2 }\). The capacities of the possible routes, in van loads per day, are shown in Fig 2. Warehouses \(W _ { 1 } , W _ { 2 }\) and \(W _ { 3 }\) have 14, 12 and 14 van loads respectively available per day and retail outlets \(R _ { 1 }\) and \(R _ { 2 }\) can accept 6 and 25 van loads respectively per day.
  29. On Diagram 1 on the answer sheet add a supersource \(W\), a supersink \(R\) and the appropriate directed arcs to obtain a single-source, single-sink capacitated network. State the minimum capacity of each arc you have added.
  30. State the maximum flow along
    1. \(W \quad W _ { 1 } \quad A \quad R _ { 1 } \quad R\),
    2. \(W W _ { 3 } \quad C \quad R _ { 2 } \quad R\).
  31. Taking your answers to part (b) as the initial flow pattern, use the labelling procedure to obtain a maximum flow through the network from \(W\) to \(R\). Show your working on Diagram 2. List each flowaugmenting route you use, together with its flow.
  32. From your final flow pattern, determine the number of van loads passing through \(B\) each day. \section*{D2 2003 (adapted for new spec)}
    1. A two person zero-sum game is represented by the following pay-off matrix for player \(A\).
    \cline { 2 - 4 } \multicolumn{1}{c|}{}B plays I\(B\) plays II\(B\) plays III
    \(A\) plays I- 325
    \(A\) plays II4- 1- 4
  33. Write down the pay off matrix for player \(B\).
  34. Formulate the game as a linear programming problem for player \(B\), writing the constraints as equalities and stating your variables clearly.
    2. (a) Explain the difference between the classical and practical travelling salesman problems.
    \includegraphics[max width=\textwidth, alt={}, center]{4f494f19-5690-4d9f-8c18-db03d41da203-06_454_857_737_1736} The network in the diagram above shows the distances, in kilometres, between eight McBurger restaurants. An inspector from head office wishes to visit each restaurant. His route should start and finish at \(A\), visit each restaurant at least once and cover a minimum distance.
  35. Obtain a minimum spanning tree for the network using Kruskal's algorithm. You should draw your tree and state the order in which the arcs were added.
  36. Use your answer to part (b) to determine an initial upper bound for the length of the route.
  37. Starting from your initial upper bound and using an appropriate method, find an upper bound which is less than 135 km . State your tour.
Edexcel D2 Q4
4. Andrew ( \(A\) ) and Barbara ( \(B\) ) play a zero-sum game. This game is represented by the following payoff matrix for Andrew. $$A \left( \begin{array} { c c c } & B &
3 & 5 & 4
1 & 4 & 2
6 & 3 & 7 \end{array} \right)$$
  1. Explain why this matrix may be reduced to $$\left( \begin{array} { l l } 3 & 5
    6 & 3 \end{array} \right) .$$
  2. Hence find the best strategy for each player and the value of the game.
Edexcel D2 Q6
6. The table below shows the distances, in km , between six towns \(A , B , C , D , E\) and \(F\).
\cline { 2 - 7 } \multicolumn{1}{c|}{}\(A\)\(B\)\(C\)\(D\)\(E\)\(F\)
\(A\)-85110175108100
\(B\)85-3817516093
\(C\)11038-14815673
\(D\)175175148-11084
\(E\)108160156110-92
\(F\)10093738492-
  1. Starting from \(A\), use Prim's algorithm to find a minimum connector and draw the minimum spanning tree. You must make your method clear by stating the order in which the arcs are selected.
    1. Using your answer to part (a) obtain an initial upper hound for the solution of the travelling salesman problem.
    2. Use a short cut to reduce the upper bound to a value less than 680 .
  3. Starting by deleting \(F\), find a lower bound for the solution of the travelling salesman problem.
Edexcel D2 Q7
7. A steel manufacturer has 3 factories \(F _ { 1 } , F _ { 2 }\) and \(F _ { 3 }\) which can produce 35,25 and 15 kilotonnes of steel per year, respectively. Three businesses \(B _ { 1 } , B _ { 2 }\) and \(B _ { 3 }\) have annual requirements of 20,25 and 30 kilotonnes respectively. The table below shows the cost \(C _ { i j }\) in appropriate units, of transporting one kilotonne of steel from factory \(F _ { i }\) to business \(B _ { j }\).
\cline { 3 - 5 } \multicolumn{2}{c|}{}Business
\cline { 3 - 5 } \multicolumn{2}{c|}{}\(B _ { 1 }\)\(B _ { 2 }\)\(B _ { 3 }\)
\multirow{3}{*}{Factory}\(F _ { 1 }\)10411
\cline { 2 - 5 }\(F _ { 2 }\)1258
\cline { 2 - 5 }\(F _ { 3 }\)967
The manufacturer wishes to transport the steel to the businesses at minimum total cost.
  1. Write down the transportation pattern obtained by using the North-West corner rule.
  2. Calculate all of the improvement indices \(I _ { i j }\), and hence show that this pattern is not optimal.
  3. Use the stepping-stone method to obtain an improved solution.
  4. Show that the transportation pattern obtained in part (c) is optimal and find its cost. \begin{figure}[h]
    \captionsetup{labelformat=empty} \caption{Figure 4} \includegraphics[alt={},max width=\textwidth]{4f494f19-5690-4d9f-8c18-db03d41da203-04_346_922_319_278}
    \end{figure} The network in Fig. 4 models a drainage system. The number on each arc indicates the capacity of that arc, in litres per second.
  5. Write down the source vertices. \begin{figure}[h]
    \captionsetup{labelformat=empty} \caption{Figure 5} \includegraphics[alt={},max width=\textwidth]{4f494f19-5690-4d9f-8c18-db03d41da203-04_341_920_900_278}
    \end{figure} Figure 5 shows a feasible flow through the same network.
  6. State the value of the feasible flow shown in Fig. 5. Taking the flow in Fig. 5 as your initial flow pattern,
  7. use the labelling procedure on Diagram 1 to find a maximum flow through this network. You should list each flow-augmenting route you use, together with its flow.
  8. Show the maximal flow on Diagram 2 and state its value.
  9. Prove that your flow is maximal.
    9. T42 Co. Ltd produces three different blends of tea, Morning, Afternoon and Evening. The teas must be processed, blended and then packed for distribution. The table below shows the time taken, in hours, for each stage of the production of a tonne of tea. It also shows the profit, in hundreds of pounds, on each tonne.
    \cline { 2 - 5 } \multicolumn{1}{c|}{}ProcessingBlendingPackingProfit (£100)
    Morning blend3124
    Afternoon blend2345
    Evening blend4233
    The total times available each week for processing, blending and packing are 35, 20 and 24 hours respectively. T42 Co. Ltd wishes to maximise the weekly profit. Let \(x , y\) and \(z\) be the number of tonnes of Morning, Afternoon and Evening blend produced each week.
  10. Formulate the above situation as a linear programming problem, listing clearly the objective function, and the constraints as inequalities. An initial Simplex tableau for the above situation is
    Basic
    variable
    		\(x\)\(y\)\(z\)\(r\)\(s\)\(t\)Value
    \(r\)32410035
    \(s\)13201020
    \(t\)24300124
    \(P\)- 4- 5- 30000
  11. Solve this linear programming problem using the Simplex algorithm. Take the most negative number in the profit row to indicate the pivot column at each stage.
    (11) T42 Co. Ltd wishes to increase its profit further and is prepared to increase the time available for processing or blending or packing or any two of these three.
  12. Use your answer to part (b) to advise the company as to which stage(s) it should increase the time available.
    (2)
    10. While solving a maximizing linear programming problem, the following tableau was obtained.
    Basic
    variable
    		\(x\)\(y\)\(z\)\(r\)\(s\)\(t\)Value
    \(r\)00\(1 \frac { 2 } { 3 }\)10\(- \frac { 1 } { 6 }\)\(\frac { 2 } { 3 }\)
    \(y\)01\(3 \frac { 1 } { 3 }\)01\(- \frac { 1 } { 3 }\)\(\frac { 1 } { 3 }\)
    \(x\)10- 30- 1\(\frac { 1 } { 2 }\)1
    \(P\)00101111
  13. Explain why this is an optimal tableau.
  14. Write down the optimal solution of this problem, stating the value of every variable.
  15. Write down the profit equation from the tableau. Use it to explain why changing the value of any of the non-basic variables will decrease the value of \(P\).
    11. A company wishes to transport its products from 3 factories \(F _ { 1 } , F _ { 2 }\) and \(F _ { 3 }\) to a single retail outlet \(R\). The capacities of the possible routes, in van loads per day, are shown in Fig. 5. \begin{figure}[h]
    \captionsetup{labelformat=empty} \caption{Figure 5} \includegraphics[alt={},max width=\textwidth]{4f494f19-5690-4d9f-8c18-db03d41da203-05_470_766_447_1695}
    \end{figure}
  16. On Diagram 1 in the answer booklet add a supersource \(S\) to obtain a capacitated network with a single source and a single sink. State the minimum capacity of each arc you have added.
    1. State the maximum flow along \(S F _ { 1 } A B R\) and \(S F _ { 3 } C R\).
    2. Show these maximum flows on Diagram 2 in the answer booklet, using numbers in circles. Taking your answer to part (b)(ii) as the initial flow pattern,
    1. use the labelling procedure to find a maximum flow from \(S\) to \(R\). Your working should be shown on Diagram 3. List each flow-augmenting route you find together with its flow.
    2. Prove that your final flow is maximal.
      12. \begin{figure}[h]
      \captionsetup{labelformat=empty} \caption{Figure 2} \includegraphics[alt={},max width=\textwidth]{4f494f19-5690-4d9f-8c18-db03d41da203-06_405_791_301_221}
      \end{figure} A company has 3 warehouses \(W _ { 1 } , W _ { 2 }\), and \(W _ { 3 }\). It needs to transport the goods stored there to 2 retail outlets \(R _ { 1 }\) and \(R _ { 2 }\). The capacities of the possible routes, in van loads per day, are shown in Fig 2. Warehouses \(W _ { 1 } , W _ { 2 }\) and \(W _ { 3 }\) have 14, 12 and 14 van loads respectively available per day and retail outlets \(R _ { 1 }\) and \(R _ { 2 }\) can accept 6 and 25 van loads respectively per day.
  19. On Diagram 1 on the answer sheet add a supersource \(W\), a supersink \(R\) and the appropriate directed arcs to obtain a single-source, single-sink capacitated network. State the minimum capacity of each arc you have added.
  20. State the maximum flow along
    1. \(W \quad W _ { 1 } \quad A \quad R _ { 1 } \quad R\),
    2. \(W W _ { 3 } \quad C \quad R _ { 2 } \quad R\).
  21. Taking your answers to part (b) as the initial flow pattern, use the labelling procedure to obtain a maximum flow through the network from \(W\) to \(R\). Show your working on Diagram 2. List each flowaugmenting route you use, together with its flow.
  22. From your final flow pattern, determine the number of van loads passing through \(B\) each day. \section*{D2 2003 (adapted for new spec)}
    1. A two person zero-sum game is represented by the following pay-off matrix for player \(A\).
    \cline { 2 - 4 } \multicolumn{1}{c|}{}B plays I\(B\) plays II\(B\) plays III
    \(A\) plays I- 325
    \(A\) plays II4- 1- 4
  23. Write down the pay off matrix for player \(B\).
  24. Formulate the game as a linear programming problem for player \(B\), writing the constraints as equalities and stating your variables clearly.
    2. (a) Explain the difference between the classical and practical travelling salesman problems.
    \includegraphics[max width=\textwidth, alt={}, center]{4f494f19-5690-4d9f-8c18-db03d41da203-06_454_857_737_1736} The network in the diagram above shows the distances, in kilometres, between eight McBurger restaurants. An inspector from head office wishes to visit each restaurant. His route should start and finish at \(A\), visit each restaurant at least once and cover a minimum distance.
  25. Obtain a minimum spanning tree for the network using Kruskal's algorithm. You should draw your tree and state the order in which the arcs were added.
  26. Use your answer to part (b) to determine an initial upper bound for the length of the route.
  27. Starting from your initial upper bound and using an appropriate method, find an upper bound which is less than 135 km . State your tour.
    3. Talkalot College holds an induction meeting for new students. The meeting consists of four talks: I (Welcome), II (Options and Facilities), III (Study Tips) and IV (Planning for Success). The four department heads, Clive, Julie, Nicky and Steve, deliver one of these talks each. The talks are delivered consecutively and there are no breaks between talks. The meeting starts at 10 a.m. and ends when all four talks have been delivered. The time, in minutes, each department head takes to deliver each talk is given in the table below.
    \cline { 2 - 5 } \multicolumn{1}{c|}{}Talk ITalk IITalk IIITalk IV
    Clive12342816
    Julie13323612
    Nicky15323214
    Steve11333610
  28. Use the Hungarian algorithm to find the earliest time that the meeting could end. You must make your method clear and show
    1. the state of the table after each stage in the algorithm,
    2. the final allocation.
  29. Modify the table so it could be used to find the latest time that the meeting could end.
    4. A two person zero-sum game is represented by the following pay-off matrix for player \(A\).
    \cline { 2 - 4 } \multicolumn{1}{c|}{}\(B\) plays I\(B\) plays II\(B\) plays III
    \(A\) plays I2- 13
    \(A\) plays II130
    \(A\) plays III01- 3
  30. Identify the play safe strategies for each player.
  31. Verify that there is no stable solution to this game.
  32. Explain why the pay-off matrix above may be reduced to
    \cline { 2 - 4 } \multicolumn{1}{c|}{}\(B\) plays I\(B\) plays II\(B\) plays III
    \(A\) plays I2- 13
    \(A\) plays II130
  33. Find the best strategy for player \(A\), and the value of the game.
    5. The manager of a car hire firm has to arrange to move cars from three garages \(A , B\) and \(C\) to three airports \(D , E\) and \(F\) so that customers can collect them. The table below shows the transportation cost of moving one car from each garage to each airport. It also shows the number of cars available in each garage and the number of cars required at each airport. The total number of cars available is equal to the total number required.
    Airport \(D\)Airport EAirport \(F\)Cars available
    Garage \(A\)£20£40£106
    Garage \(B\)£20£30£405
    Garage \(C\)£10£20£308
    Cars required694
  34. Use the North-West corner rule to obtain a possible pattern of distribution and find its cost.
  35. Calculate shadow costs for this pattern and hence obtain improvement indices for each route.
  36. Use the stepping-stone method to obtain an optimal solution and state its cost.
    6. Kris produces custom made racing cycles. She can produce up to four cycles each month, but if she wishes to produce more than three in any one month she has to hire additional help at a cost of \(\pounds 350\) for that month. In any month when cycles are produced, the overhead costs are \(\pounds 200\). A maximum of 3 cycles can be held in stock in any one month, at a cost of \(\pounds 40\) per cycle per month. Cycles must be delivered at the end of the month. The order book for cycles is
    MonthAugustSeptemberOctoberNovember
    Number of cycles required3352
    Disregarding the cost of parts and Kris' time,
  37. determine the total cost of storing 2 cycles and producing 4 cycles in a given month, making your calculations clear. There is no stock at the beginning of August and Kris plans to have no stock after the November delivery.
  38. Use dynamic programming to determine the production schedule which minimises the costs, showing your working in the table below.
    StageDemandStateActionDestinationValue
    \multirow[t]{3}{*}{1 (Nov)}\multirow[t]{3}{*}{2}0 (in stock)(make) 20200
    1 (in stock)(make) 10240
    2 (in stock)(make) 0080
    \multirow[t]{2}{*}{2 (Oct)}\multirow[t]{2}{*}{5}140\(590 + 200 = 790\)
    230
    The fixed cost of parts is \(\pounds 600\) per cycle and of Kris' time is \(\pounds 500\) per month. She sells the cycles for \(\pounds 2000\) each.
  39. Determine her total profit for the four month period.
    (Total 18 marks)
  40. Find the value of cuts \(C _ { 1 }\) and \(C _ { 2 }\). Starting with the given feasible flow of 68,
  41. use the labelling procedure on Diagram 2 to find a maximal flow through this network. List each flow-augmenting route you use, together with its flow.
    7. \begin{figure}[h]
    \captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{4f494f19-5690-4d9f-8c18-db03d41da203-08_499_1011_536_246}
    \end{figure} Figure 1 shows a capacitated, directed network. The unbracketed number on each arc indicates the capacity of that arc, and the numbers in brackets show a feasible flow of value 68 through the network.
  42. Add a supersource and a supersink, and arcs of appropriate capacity, to Diagram 1 below. \section*{Diagram 1} \begin{figure}[h]
    \captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{4f494f19-5690-4d9f-8c18-db03d41da203-08_451_1013_1274_246}
    \end{figure}
  43. Find the values of \(x\) and \(y\), explaining your method briefly. \section*{Diagram 2} \includegraphics[max width=\textwidth, alt={}, center]{4f494f19-5690-4d9f-8c18-db03d41da203-08_438_1010_463_1663}
  44. Show your maximal flow on Diagram 3 and state its value. \section*{Diagram 3} \includegraphics[max width=\textwidth, alt={}, center]{4f494f19-5690-4d9f-8c18-db03d41da203-08_437_1006_1105_1665}
  45. Prove that your flow is maximal.
Edexcel D2 Q9
9. T42 Co. Ltd produces three different blends of tea, Morning, Afternoon and Evening. The teas must be processed, blended and then packed for distribution. The table below shows the time taken, in hours, for each stage of the production of a tonne of tea. It also shows the profit, in hundreds of pounds, on each tonne.
\cline { 2 - 5 } \multicolumn{1}{c|}{}ProcessingBlendingPackingProfit (£100)
Morning blend3124
Afternoon blend2345
Evening blend4233
The total times available each week for processing, blending and packing are 35, 20 and 24 hours respectively. T42 Co. Ltd wishes to maximise the weekly profit. Let \(x , y\) and \(z\) be the number of tonnes of Morning, Afternoon and Evening blend produced each week.
  1. Formulate the above situation as a linear programming problem, listing clearly the objective function, and the constraints as inequalities. An initial Simplex tableau for the above situation is
    Basic
    variable
    		\(x\)\(y\)\(z\)\(r\)\(s\)\(t\)Value
    \(r\)32410035
    \(s\)13201020
    \(t\)24300124
    \(P\)- 4- 5- 30000
  2. Solve this linear programming problem using the Simplex algorithm. Take the most negative number in the profit row to indicate the pivot column at each stage.
    (11) T42 Co. Ltd wishes to increase its profit further and is prepared to increase the time available for processing or blending or packing or any two of these three.
  3. Use your answer to part (b) to advise the company as to which stage(s) it should increase the time available.
    (2)
Edexcel D2 Q10
10. While solving a maximizing linear programming problem, the following tableau was obtained.
Basic
variable
\(x\)\(y\)\(z\)\(r\)\(s\)\(t\)Value
\(r\)00\(1 \frac { 2 } { 3 }\)10\(- \frac { 1 } { 6 }\)\(\frac { 2 } { 3 }\)
\(y\)01\(3 \frac { 1 } { 3 }\)01\(- \frac { 1 } { 3 }\)\(\frac { 1 } { 3 }\)
\(x\)10- 30- 1\(\frac { 1 } { 2 }\)1
\(P\)00101111
  1. Explain why this is an optimal tableau.
  2. Write down the optimal solution of this problem, stating the value of every variable.
  3. Write down the profit equation from the tableau. Use it to explain why changing the value of any of the non-basic variables will decrease the value of \(P\).
Edexcel D2 Q11
11. A company wishes to transport its products from 3 factories \(F _ { 1 } , F _ { 2 }\) and \(F _ { 3 }\) to a single retail outlet \(R\). The capacities of the possible routes, in van loads per day, are shown in Fig. 5. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 5} \includegraphics[alt={},max width=\textwidth]{4f494f19-5690-4d9f-8c18-db03d41da203-05_470_766_447_1695}
\end{figure}
  1. On Diagram 1 in the answer booklet add a supersource \(S\) to obtain a capacitated network with a single source and a single sink. State the minimum capacity of each arc you have added.
    1. State the maximum flow along \(S F _ { 1 } A B R\) and \(S F _ { 3 } C R\).
    2. Show these maximum flows on Diagram 2 in the answer booklet, using numbers in circles. Taking your answer to part (b)(ii) as the initial flow pattern,
    1. use the labelling procedure to find a maximum flow from \(S\) to \(R\). Your working should be shown on Diagram 3. List each flow-augmenting route you find together with its flow.
    2. Prove that your final flow is maximal.